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A291717
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Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1 <= k <= m positions can be picked in an m X m square grid such that the picked positions have a central symmetry.
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10
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1, 4, 6, 9, 36, 8, 16, 120, 24, 168, 25, 300, 72, 714, 178, 36, 630, 144, 2273, 464, 6576, 49, 1176, 288, 5932, 1476, 24288, 6404, 64, 2016, 480, 13536, 3040, 74560, 15680, 341320, 81, 3240, 800, 27860, 6940, 197600, 50860, 1170466, 314862
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OFFSET
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1,2
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LINKS
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EXAMPLE
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A configuration of 6 picked points from a 7 X 7 grid with a central (point) symmetry w.r.t. point #, but no line (mirror) symmetry and thus only contributing to T(7,6)=a(27), but not to A291718(27), would be:
o o o X o o o
o o o o o o o
o o o o X o o
o X # X o o o
X o o o o o o
o o o o o o o
o X o o o o o
.
Triangle begins:
1;
4, 6;
9, 36, 8;
16, 120, 24, 168;
25, 300, 72, 714, 178;
36, 630, 144, 2273, 464, 6576;
49, 1176, 288, 5932, 1476, 24288, 6404;
64, 2016, 480, 13536, 3040, 74560, 15680, 341320;
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MATHEMATICA
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decentralize[v_] := 2*Total[v] - Last[v];
T[n_, k_] := decentralize[ Table[ decentralize[ Table[ If[EvenQ[k] || OddQ[a*b], Binomial[ Quotient[a*b, 2], Quotient[k, 2]], 0], {b, 1, n}]], {a, 1, n}]];
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PROG
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(PARI)
decentralize(v) = 2*vecsum(v) - v[length(v)];
T(n, k) = decentralize(vector(n, a, decentralize(vector(n, b, if(k%2==0||a*b%2==1, binomial(a*b\2, k\2))))));
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Sep 16 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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